The functions f and g are such that
f(x) = x - 8
g(x) = x2 + 6x
The function h is such that h(x) = fg(x)
The domain of h is {x : x ≥ -3}
Express the inverse function h-1 in the form h-^ (x) = ...

Given the functions \( f(x) = x - 8 \) and \( g(x) = x^2 + 6x \), we need to find the function \( h \) such that \( h(x) = f(g(x)) \), and then express the inverse function \( h^{-1}(x) \).

First, let's find \( h(x) \):
\[ h(x) = f(g(x)) \]
\[ g(x) = x^2 + 6x \]
\[ f(x) = x - 8 \]

So,
\[ h(x) = f(g(x)) = f(x^2 + 6x) = (x^2 + 6x) - 8 \]
\[ h(x) = x^2 + 6x - 8 \]

Next, we need to find the inverse function \( h^{-1}(x) \). To do this, we solve the equation \( y = h(x) \) for \( x \) in terms of \( y \).

\[ y = x^2 + 6x - 8 \]

Rearrange the equation:
\[ x^2 + 6x - (y + 8) = 0 \]

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \). We can solve for \( x \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = 6 \), and \( c = -(y + 8) \).

\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-(y + 8))}}{2 \cdot 1} \]
\[ x = \frac{-6 \pm \sqrt{36 + 4(y + 8)}}{2} \]
\[ x = \frac{-6 \pm \sqrt{36 + 4y + 32}}{2} \]
\[ x = \frac{-6 \pm \sqrt{68 + 4y}}{2} \]
\[ x = \frac{-6 \pm 2\sqrt{17 + y}}{2} \]
\[ x = -3 \pm \sqrt{17 + y} \]

Since the domain of \( h \) is \( \{x : x \geq -3\} \), we need to select the positive branch of the solution:
\[ x = -3 + \sqrt{17 + y} \]

Thus, the inverse function \( h^{-1}(x) \) is:
\[ h^{-1}(x) = -3 + \sqrt{17 + x} \]

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How do you find the domain of a composite function?
2. What is the quadratic formula and how is it derived?
3. How do you verify if a function is invertible?
4. Can you provide an example of another composite function and its inverse?
5. What are the properties of inverse functions?

**Tip:** When solving for the inverse of a function, always check the domain and range to ensure the inverse is properly defined.

Learn how to find the inverse function h^-1(x) for composite functions f(x) = x - 8 and g(x) = x^2 + 6x. Understand the process of solving for the inverse function and applying the quadratic formula. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation